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%% lecture06.tex
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%% Started on  Thu Jan  5 08:04:52 2012 alex
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\exercises
\begin{xca}\index{Klein Bottle}
Describe a cell decomposition of Klein bottle and use it to
calculate homology groups with integer coefficients and with
coefficients in cyclic group $\ZZ_k$.

Calculate the fundamental group of Klein bottle and check that
its abelianization gives one-dimensional homology with integer
coefficients.
\end{xca}
\begin{xca}
Describe cell decompositions of the following spaces. Calculate
homology groups with integer coefficients and with coefficients
in cyclic group $\ZZ_k$.
\begin{enumerate}
\item The quotient of $S^2$ obtained by identifying north and
  south poles to a point.
\item The quotient of $S^2$ under identification of $x$ with $-x$
  for $x$ in the equator $S^1$.
\item The quotient space of $S^1\times S^1$ obtained by
  identifying points in the circle $S^1\times\{x_0\}$ that differ
  by $2\pi/m$ rotation and identifying points in the circle
  $\{x_0\}\times S^1$ that differ by $2\pi/n$ rotation.
\end{enumerate}
\end{xca}
